The binarity of the local white dwarf population

(1)

DOI: 10.1051 /0004-6361/201629978 c

ESO 2017

Astronomy

&

Astrophysics

The binarity of the local white dwarf population

S. Toonen

^{1, 2, 3}

, M. Hollands

⁴

, B. T. Gänsicke

⁴

, and T. Boekholt

^{2, 5}

1

Anton Pannekoek Institute for Astronomy, University of Amsterdam, 1090 GE Amsterdam, The Netherlands e-mail: toonen@strw.leidenuniv.nl

2

Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands

3

Department of Astrophysics, IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands

4

Department of Physics, University of Warwick, Coventry CV4 7AL, UK

5

Departamento de Astronomía, Facultad Ciencias Físicas y Matemáticas, Universidad de Concepción, Av. Esteban Iturra s /n Barrio Universitario, Casilla 160, Concepción, Chile

Received 30 October 2016 / Accepted 20 February 2017

ABSTRACT

Context.

As endpoints of stellar evolution, white dwarfs (WDs) are powerful tools to study the evolutionary history of the Galaxy. In particular, the multiplicity of WDs contains information regarding the formation and evolution of binary systems.

Aims.

Can we understand the multiplicity of the local WD sample from a theoretical point of view? Population synthesis methods are often applied to estimate stellar space densities and event rates, but how well are these estimates calibrated? This can be tested by a comparison with the 20 pc sample, which contains '100 stars and is minimally affected by selection biases.

Methods.

We model the formation and evolution of single stars and binaries within 20 pc with a population synthesis approach. We construct a model of the current sample of WDs and differentiate between WDs in different configurations, that is single WDs, and resolved and unresolved binaries containing a WD with either a main-sequence (MS) component or with a second WD. We also study the effect of different assumptions concerning the star formation history, binary evolution, and the initial distributions of binary parameters. We compile from the literature the available information on the sample of WDs within 20 pc, with a particular emphasis on their multiplicity, and compare this to the synthetic models.

Results.

The observed space densities of single and binary WDs are well reproduced by the models. The space densities of the most common WD systems (single WDs and unresolved WD-MS binaries) are consistent within a factor two with the observed value. We find a discrepancy only for the space density of resolved double WDs. We exclude that observational selection effects, fast stellar winds, or dynamical interactions with other objects in the Milky Way explain this discrepancy. We find that either the initial mass ratio distribution in the solar neighbourhood is biased towards low mass-ratios, or more than ten resolved DWDs have been missed observationally in the 20 pc sample. Furthermore, we show that the low binary fraction of WD systems (∼25%) compared to solar- type MS-MS binaries (∼50%) is consistent with theory, and is mainly caused by mergers in binary systems, and to a lesser degree by WDs hiding in the glare of their companion stars. Lastly, Gaia will dramatically increase the size of the volume-limited WD sample, detecting the coolest and oldest WDs out to '50 pc. We provide a detailed estimate of the number of single and binary WDs in the Gaia sample.

Key words.

binaries: close – white dwarfs – stars: evolution

1. Introduction

As most stars end their life as white dwarfs (WDs), they form a significant component of the stellar population and are the most common stellar remnants. As such, WD stars play an im- portant role in the study of the structure and the evolutionary history of stellar ensembles (Fontaine et al. 2001; Althaus et al.

2010). They provide us with an e ffective way to reconstruct the star formation history (SFH) of the solar neighbourhood and Galactic disc by analyzing the WD luminosity function (e.g. Tremblay et al. 2014). WDs can also be used to con- strain with good accuracy the age of stellar ensembles, such as the solar neighbourhood, stellar clusters, and the Galactic disc (Torres et al. 2005; Hansen et al. 2007; Bedin et al. 2009). Fun- damental for these types of studies are observational samples that are as large and homogeneously-selected as possible.

An important, but often complicated aspect in many pop- ulation studies, is the level of completeness of the observa- tional sample and how to compensate for any observational biases. A complete sample of WDs is therefore a powerful

tool, but assembling such a sample can be observationally very demanding, as WDs are low-luminosity objects, and the dif- ferent WD discovery methods, primarily proper motion sur- veys and ultraviolet excess surveys, have incomplete over- lap. Much time and e ffort has been devoted to create a com- plete and volume-limited sample of WDs in the solar neigh- bourhood (e.g. Holberg et al. 2002, 2008b; Vennes & Kawka 2003; Kawka et al. 2004; Kawka & Vennes 2006; Farihi et al.

2005; Subasavage et al. 2007, 2008; Sion et al. 2009, 2014;

Giammichele et al. 2012; Sayres et al. 2012; Limoges et al.

2013, 2015). The advantage of the solar neighbourhood is that even the coolest WDs can be identified with relative ease at these short distances from us (e.g. Carrasco et al. 2014).

The level of completeness that has been achieved for the WD sample within 20 pc is exceptional, and is estimated to be 80–90% (Holberg et al. 2008b, 2016; Sion et al. 2009;

Giammichele et al. 2012).

Large and homogeneously-selected samples of stellar sys-

tems play a vital role in the empirical verification of population

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synthesis studies, such as binary population synthesis (BPS)

¹

. The BPS approach aims to further improve our understanding of stellar and binary evolution from a statistical point of view, and can aid and further motivate observational surveys. It is often used to constrain evolutionary pathways and predict population characteristics, such as event rates or the period distribution of stellar populations, including type Ia supernovae (for a review see Wang & Han 2012), post-common envelope binaries (e.g. Toonen & Nelemans 2013; Camacho et al. 2014;

Zorotovic et al. 2014), or AM CVn systems (e.g. Nelemans et al.

2001a). Nonetheless, tests on the number densities of a stellar population (e.g. space densities or event rates) predicted by BPS studies are often not strongly constraining, as the observed num- ber densities are uncertain to (at least) a factor of a few. However, since the 20 pc sample of WDs is volume-limited and nearly complete, it allows for a strong test of the number of predicted systems from the BPS method, which is the aim of this paper.

Another important feature of the 20 pc sample is that it con- sists of multiple populations of WD systems. It contains WDs formed by single stellar evolution and from mergers in bina- ries, and WDs in binaries such as double WDs (DWDs) and WD main-sequence binaries (WDMS). The sample provides us with a rare opportunity to compare multiple stellar populations, formed from very di fferent evolutionary paths, with the results of self-consistent population synthesis models. So far, none of the studies of the WD luminosity function have included bina- rity (e.g. Tremblay et al. 2014; Torres & García-Berro 2016), de- spite the expected contribution from binaries (van Oirschot et al.

2014).

The set-up of this paper is as follows: in Sect. 2, we give an overview of the observed sample of local WDs. In Sect. 3, we describe the BPS simulations. In Sect. 4 the self-consistent sim- ulated WD populations are presented. We compare the number of systems in the WD population and its subcomponents pre- dicted by the synthetic populations with the observed sample of Sect. 2. For unresolved binaries, we take into account the selec- tion e ffects against finding a dim star next to a bright star. We also predict the number of WD systems within 50 pc in Sect. 5, which will become available with Gaia. In Sect. 6 we discuss the hypothesis of missing WD binaries in the solar neighbourhood, and in Sect. 7 our results are summarized.

2. Observed sample

Holberg et al. (2002) constructed a local WD sample consist- ing of 109 WD candidates within 20 pc. Holberg et al. (2002) estimated that their sample was approximately 65% complete.

Since then the completeness of the local WD sample was esti- mated to have risen to 80–90% (Holberg et al. 2008b; Sion et al.

2009; Giammichele et al. 2012). Most recently, the complete- ness level has been estimated to be 86% by Holberg et al.

(2016). The local WD sample has been used to derive the lo- cal space density [(4.8 ± 0.5) × 10

⁻³

pc

⁻³

] and mass density [(3.1 ± 0.3) × 10

⁻³

M pc

⁻³

] (e.g. Holberg et al. 2002, 2008b, 2016; Sion et al. 2009). The kinematical properties of the lo- cal WD sample have been studied by Sion et al. (2009), who found that the vast majority of these stars belong to the thin disk. Finally, Giammichele et al. (2012) performed a systematic model atmosphere analysis of all the available data of the local WD population.

The observed sample that we use here is mainly based on the sample of systems from Giammichele et al. (2012) and full de- tails are given in Appendix A and Table A.1. The sample of WDs

1

See Toonen et al. (2014) for a comparison of four BPS codes.

in binaries is given in Table 1, and WDs in higher-order systems in Table 2. A good starting point on WD binarity is provided by Farihi et al. (2005), Holberg et al. (2008b, 2013). We note that the latter paper focuses on Sirius-type binaries (WDs with com- panions of spectral K and earlier) in the solar neighbourhood, but is incomplete with respect to low-mass companions. Notes on specific WD systems are given below.

2.1. Notes on individual objects

2.1.1. A new resolved double degenerate at 33 pc

We report the identification of a new resolved double degen- erate system, comprising WD0648 +641 and the recently dis- covered WD0649 +639. The two stars are 8.2 arcmin apart and their proper motions are (432, –142) mas/yr and (421, –130) mas /yr, respectively ( Lépine & Shara 2005). The trigono- metric distance to WD0648 +641 has been determined to be 33 ± 5 pc (van Altena et al. 1995), and the spectroscopic distance to WD0649 +639 is about 21 pc ( Limoges et al. 2013, 2015).

Nevertheless, since the temperatures, spectroscopic masses, and V-band magnitudes of both WDs are very comparable (6220 ± 137 K versus 6050 ± 98 K, 0.87 ± 0.15 M versus 0.98 ± 0.09 M , and 14.67 versus 15.07 for WD0649 +639 and WD0648+641, respectively, see Limoges et al. 2015), we deem it likely that the two WDs are at a comparable distance.

2.1.2. Distances

The distances given in Table 1 are based on Giammichele et al.

(2012) with updates from Limoges et al. (2013), Limoges et al.

(2015), and the Discovery and Evaluation of Nearby Stellar Em- bers (DENSE) project

²

. For a few systems, the derived distances from di fferent studies are significantly discrepant, such that their membership of the 20 pc sample is ambiguous. We discuss these systems here in detail.

– WD0019 +423 has a spectroscopic distance of 12.9 ± 3.0 pc (Limoges et al. 2015). However, its V-band magnitude of 16.5, e ffective temperature of 5590 K, and log g of 8.0 from Limoges et al. (2015) implies an absolute magnitude of 14.5 (using the WD models as described in Sect. 3.6) and a dis- tance of 25 pc. This system is therefore removed from the 20 pc sample.

– WD0454 +620 is an unresolved WDMS system in which the M-dwarf contaminates the WD spectrum. Both Limoges et al. (2013) and Limoges et al. (2015) take special care in the fitting procedure of the WD spectral lines, how- ever, the derived distances are distinct. The most recent mea- surement of Limoges et al. (2015) gives a distance of 21.6 ± 1.2 pc, which gives a 10% chance for the system to be within 20 pc. With the distance found by Limoges et al. (2013) (24.9 ± 0.9 pc) it is excluded that WD0454 +620 is within 20 pc. We adopt the most recent value of Limoges et al.

(2015), however, we note that this does not significantly af- fect our conclusions of Sect. 7.

– WD1242−105 has recently been shown not to be a sin- gle object, but to be part of a double degenerate binary (Debes et al. 2015) with a short period of 2.85 h. These au- thors find a trigonometric distance of 39 ± 1 pc, which ex- cludes WD1242−105 from the 20 pc sample. Previously, the distance to WD1242−105 was estimated to be 23.5 ± 1 pc (Giammichele et al. 2012), based on spectral model fitting assuming a single object.

2

http://www.DenseProject.com

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Table 1. Known WDs in binary systems in the solar neighbourhood.

WD name Distance [pc] Spectral Mass [M ] log L /L Companion Spectral Angular References

type name type separation [

⁰⁰

]

Resolved WDMS

0148 +641 17.35 (0.15) DA5.6 0.66 (0.03) –3.08 GJ 3117 A M2 12.1 1, 2, 3, 4, 5

0208−510 10.782 (0.004) DA6.9 0.59 (0.01) – GJ86A K0 1.9 3, 6, 7

0415−594 18.46 (0.05) DA3.3 0.60 (0.02) – eps. Reticulum A K2 12.8 7, 9,10

0426 +588 5.51 (0.02) DC7.1 0.69 (0.02) –3.52 GJ 169.1 A M4.0 9.2 4, 5, 8, 11, 12, 13, 14

0628−020 20.49 (0.46) DA7.2 0.62 (0.01) – LDS 5677B M 4.5 15, 16, 17

0642−166 2.631 (0.009) DA2.0 0.98 (0.03) –1.53 Sirius A A0 7.5 8, 10, 18

0736 +053 3.50 (0.01) DQZ6.5 0.63 (0.00) –3.31 Procyon A F5 53 10, 19

0738−172 9.096 (0.046) DZA6.6 0.62 (0.02) –3.35 GJ 238 B M6.5 21.4 3, 8, 20, 21

0751−252 17.78 (0.13) DA9.9 0.59 (0.02) –4.02 LTT2976 M0 400 3, 4, 5, 22, 23

1009−184 18.3 (0.3) DZ8.3 0.59 (0.02) –3.74 LHS 2031 A K7 400 3, 10, 24, 25, 26

1043−188 19.01 (0.18) DQpec8.7 0.53 (0.11) –3.77 GJ 401 A M3 8 3, 4, 5, 15

1105−048 17.33 (3.75) DA3.5 0.54 (0.01) – LP 672–2 M3 279 2, 17, 27, 28

1132−325 9.560 (0.034) DC10 – – HD 100623 K0 16 8, 10, 17, 29

1327−083 16.2 (0.7) DA3.5 0.61 (0.03) –2.16 LHS 353 M4.5 503 3, 28, 30, 31

1345 +238 12.1 (0.3) DC11.0 0.45 (0.02) –4.08 LHS 362 M5 199 3, 31, 32

1544−377 15.25 (0.12) DA4.8 0.55 (0.03) –2.67 GJ 599 A G6 15.2 3, 4, 5, 8, 10, 33 1620−391 12.792 (0.062) DA2.1 0.61 (0.02) –1.12 HD 147513 G5 345 8, 10, 27, 31, 34

1917−077 10.1 (0.3) DBQA4.8 0.62 (0.02) –2.81 LDS 678B M6 27.3 3, 20, 31

2011 +065 22.4 (1.0) DC7.6 0.7 (0.04) –3.68 LHS 3533 M3.5 101 13, 26, 35

2151−015 24.5 (1.0) DA5.5 0.58 (0.03) –2.96 LTT 8747B M8 1.082 3, 31, 36

2154−512 15.12 (0.12) DQP8.3 0.60 (0.04) –3.44 GJ841 A M2 28.5 3, 4, 5, 30, 37, 38

2307 +548 16.2 (0.7) DA8.8 0.58 (–) – G233–42 M5 6 13, 17, 39, 40

2307−691 20.94 (0.38) DA5 0.57 (–) – GJ 1280 K3 13.1 17

2341+322 17.61 (0.55) DA4.0 0.56 (0.03) –2.3 G130–6 M3 175 20, 41, 42

Unresolved WDMS

0419−487 20.13 (0.55) DA7.8 0.22 (0.05) –3.14 – M4 P = 0.3037 3, 16, 43, 44, 45

0454 +620 21.6 (1.2) DA4.6 1.14 (0.07) – – – – 13, 39

Resolved DWD

0648 +641 33.3 (5.9) DA8.3 0.98 (0.09) –4.09 WD0649 +639 DA8.1 490 13, 28, 46, this work 0747 +073A 18.3 (0.2) DC10.4 0.48 (0.01) –4.20 WD0747 +073B DC12 16.4 27, 47

2126 +734 21.2 (0.8) DA3.2 0.60 (0.03) –1.97 – DC10 1.4 13, 31, 48

2226−754 13.5 (0.9) DC12.0 0.58 (0.00) –4.32 WD2226−755 DC12.0 93 3, 49

Unresolved DWD

0135−052 12.3 (0.4) DA6.9 0.24 (0.01) –3.00 – DA6.9 P = 1.56 27, 50

0532 +414 22.4 (1.0) DA6.5 0.52 (0.03) –3.20 – – – 3, 49

Unresolved DWD candidate

0108+277 28.0 (1.5) DA7.8 0.59 (0.00) –3.60 – – – 3

0121−429 18.3 (0.3) DAH8.0 0.41 (0.01) –3.46 – – – 3

0423 +120 17.4 (0.8) DC8.2 0.65 (0.04) –3.75 – – – 3, 25

0503−174 21.9 (1.9) DAH9.5 0.38 (0.07) –3.75 – – – 3

0839−327 8.80 (0.15) DA5.6 0.44 (0.07) –2.84 – – – 3, 8

2048 +263 20.1 (1.4) DA9.9 0.24 (0.04) –3.65 – – – 3

2248 +293 20.9 (1.9) DA9.0 0.35 (0.07) –3.62 – – – 3

2322+137 22.3 (1.0) DA9.7 0.35 (0.03) –3.75 – – – 3

Notes. The distances, spectral types, masses, and luminosities are taken from Giammichele et al. (2012). References for the binarity of the system are given in the last column. For the unresolved systems, the period P is given in days instead of angular separation, if available.

¹

Greenstein (1970);

²

Wegner (1981);

³

Giammichele et al. (2012);

⁴

Tremblay et al. (2017);

⁵

Gaia Collaboration (2016);

⁶

Mugrauer & Neuhäuser (2005);

⁷

van Leeuwen (2007);

⁸

http://www.DenseProject.com;

⁹

Farihi et al. (2011a);

¹⁰

Holberg et al. (2013);

¹¹

Liebert (1976);

12

Heintz (1990);

¹³

Limoges et al. (2015);

¹⁴

Dieterich et al. (2012);

¹⁵

Oswalt et al. (1988);

¹⁶

Subasavage et al. (2009);

¹⁷

Holberg et al.

(2016);

¹⁸

Gatewood & Gatewood (1978);

¹⁹

Liebert et al. (2013);

²⁰

Luyten (1949);

²¹

Davison et al. (2015);

²²

Subasavage et al. (2008);

23

Luyten & Hughes (1980);

²⁴

Henry et al. (2002);

²⁵

Holberg et al. (2008b);

²⁶

Hawley et al. (1996);

²⁷

Sion et al. (2014);

²⁸

van Altena et al.

(1995);

²⁹

Poveda et al. (1994);

³⁰

Eggen (1956);

³¹

Farihi et al. (2005);

³²

Dahn & Harrington (1976);

³³

Wegner (1973);

³⁴

Alexander & Lourens (1969);

³⁵

Giclas et al. (1959);

³⁶

Farihi et al. (2006);

³⁷

Vornanen et al. (2010);

³⁸

Tamazian & Malkov (2014);

³⁹

Limoges et al. (2013);

40

Newton et al. (2014);

⁴¹

Sion & Oswalt (1988);

⁴²

Garcés et al. (2011);

⁴³

Bessell & Wickramasinghe (1979);

⁴⁴

Bruch & Diaz (1998);

45

Maxted et al. (2007);

⁴⁶

Lépine & Shara (2005);

⁴⁷

Greenstein (1970);

⁴⁸

Zuckerman et al. (1997);

⁴⁹

Scholz et al. (2002);

⁵⁰

Saffer et al. (1998)

51

Zuckerman et al. (2003).

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Table 2. Known WDs in the solar neighbourhood that are part of triples and quadruples.

Distance [pc] Spectral Mass [M ] log L/L Companion Spectral Angular References

type name type separation [

⁰⁰

]

0101+048 21.3 (1.7) DA6.3 0.36 (0.05) –2.96 – DC see text 1, 2, 3, 4

HD 6101 K3 +K8 1276

0326−273 17.4 (4.3) DA5.9 0.45 (0.18) –2.97 – DC8 P = 1.88d 4, 5, 6

GB 1060B M3.5 7

0413−077 4.984 (0.006) DA3.1 0.59 (0.03) –1.85 40 Eri A K0.5 83.4 7, 8, 9, 10

40 Eri C M4.5 11.9

0433 +270 17.48 (0.13) DA9 0.62 (0.02) –3.87 V833 Tau K2

¹⁴

123.9 4, 8, 11, 12, 13, 14 0727+482A 11.1 (0.1) DC10 0.51 (0.01) –4.01 WD0727+482B DC10.1 0.656 9, 15, 16, 17

G107–69 M5

^∗

103.2 0743−336 15.2 (0.1) DC10.6 0.55 (0.01) –4.23 171 Pup A F9 870 8, 18, 19, 20

1633 +572 14.4 (0.5) DQpec8.1 0.57 (0.04) –3.75 CM draconis M4.5

^∗

26 20, 21 2054−050 16.09 (0.14) DC11.6 0.37 (0.06) –4.11 Ross 193 M3.0 15.1 4, 10, 11, 12, 22, 23 2351−335 22.90 (0.75)

^9,25

DA5.7 0.58 (0.03) –3.03 LDS826B M3.5 6.6 4, 6, 24, 25, 26

LDS826C M8.5 103

Notes. The distances, spectral types, masses, and luminosities are taken from Giammichele et al. (2012). For the unresolved systems, the period P is given instead of angular separation.

¹

Saffer et al. (1998);

²

Maxted et al. (2000);

³

Caballero (2009);

⁴

Giammichele et al. (2012);

⁵

Nelemans et al.

(2005);

⁶

Luyten (1949);

⁷

Holberg et al. (2012);

⁸

Holberg et al. (2013);

⁹

Sion et al. (2014);

¹⁰

Discovery and Evaluation of Nearby Stel- lar Embers (DENSE) project, http://www.DenseProject.com;

¹¹

Tremblay et al. (2017);

¹²

Gaia Collaboration (2016);

¹³

Hartmann et al.

(1981);

¹⁴

Tokovinin et al. (2006);

¹⁵

Strand et al. (1976);

¹⁶

Harrington et al. (1981);

¹⁷

Buscombe & Foster (1998);

¹⁸

Hartkopf et al. (2012);

19

Tokovinin et al. (2012);

²⁰

Limoges et al. (2015);

²¹

Morales et al. (2009);

²²

van Biesbroeck (1961);

²³

Tamazian & Malkov (2014);

24

Scholz et al. (2004);

²⁵

Farihi et al. (2005);

²⁶

Subasavage et al. (2009);

^(∗)

Spectral type corresponds to an unresolved binary.

– Regarding WD1657 +321, Giammichele et al. (2012) find a distance of >50 pc when assuming a log g of 8.0. On the other hand, Kawka & Vennes (2006) derive log g = 8.76 ± 0.20 and a distance d = 22 pc. Kawka & Vennes ( 2006) do not provide an uncertainty on the distance. We tenta- tively assume an uncertainty of ±1 pc, which gives a 3%

probability for WD1657 +321 to be within 20 pc. Even with an uncertainty of 2 pc on the distance estimate of Kawka & Vennes (2006, and subsequently a probability of 20% of being a member of the 20 pc sample), the space den- sity within 20 pc does not change in a significant way.

– For WD1912 +143, we adopt the trigonometric distance 35 ± 6.6 pc (Dahn et al. 1982; Limoges et al. 2015), which e ffectively excludes it from the 20 pc sample. This value is in agreement with the trigonometric distance found by van Altena et al. (1995) of 36.2 ± 7.5 pc, significantly ex- ceeding the spectroscopic distance found by Limoges et al.

(2013) of 19.4 ± 0.7 pc.

– WD2011+065 has a trigonometric distance of 22.4 ± 1.0 pc based on the parallax measurement of 44.7 ± 1.9 mas (van Altena et al. 1995; Bergeron et al. 1997). No- tably, Limoges et al. (2015) find a larger uncertainty on the distance (2.4 pc) based on the same parallax measurement.

In the former case, there is a ∼1% chance that WD2011 +065 falls within 20 pc, whereas an uncertainty of 2.4 pc gives a chance of about 15%. In both cases, WD2011 +065 does not significantly contribute to the space density within 20 pc.

– WD2151−015 is part of a binary with a MS compan- ion (Farihi et al. 2005, 2006; Holberg et al. 2008b). The binary has been resolved with an angular separation of 1.082 ± 0.002

⁰⁰

(Farihi et al. 2006). The distance found by Giammichele et al. (2012) of 24.5 ± 1.0 pc places it well out- side 20 pc, however, other estimates place it on the bound- ary of the 20 pc sample, for example 21 pc by Farihi et al.

(2006) and 20.97 ± 1.21 pc by Holberg et al. (2008b). The latter gives a 20% probability for the system to be within 20 pc.

2.1.3. Double WD candidates

A number of systems are classified as (unresolved) DWD candi- dates in Table 1. These are:

– WD0423 +120 which is overly bright for its parallax (Holberg et al. 2008b) and therefore considered to be a DWD candidate by these authors. Both the parallax and photomet- ric distances (17.36 pc vs. 11.88 pc, respectively), position the system within 20 pc from the Sun.

– WD0839−327 which is classified as a DWD candidate due to possible radial variations in the DA star (Bragaglia et al.

1990). This claim is supported by the marginal di fference in the photometric and trigonometric distance (7 pc and 8.87 ± 0.77 pc respectively) found by Kawka et al. (2007).

The trigonometric distance as given by DENSE is 8.80 ± 0.15 pc (see Table 1). Holberg et al. (2008a) found a photo- metric distance of 8.05 ± 0.11 pc.

– WD2048 +263 which is suspected to be a double-degenerate system by Bergeron et al. (2001) based on the low-gravity and mass, as well as the suspected dilution of the Balmer Hα profile of the visible DA WD by a possible DC companion.

– WD0108 +277, WD0121−429, WD0839−327, WD0503–

174, WD2054−050, and WD2248+293 which are suggested to be double degenerates by Giammichele et al. (2012). This is based on the low mass they derive by means of the pho- tometric technique. The masses are too low for stars to have evolved as single stars ( .0.5 M ). For the same reason we add WD2322 +137, however, it has a low probability of being within 20 pc (i.e. 1%). If WD2054−050 is indeed a DWD, then the system would be a triple system with an MS com- panion in a wide orbit (Greenstein 1986b,a; Sion & Oswalt 1988; Holberg et al. 2008b).

– WD0322−019 has been considered a close DWD in the past, however, Farihi et al. (2011b) showed that the source of line broadening was magnetism and not binarity.

A word of caution is necessary for the mass estimates of WDs

in unresolved binaries (and candidates). The mass estimates

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in Tables 1 and 2 are taken from Giammichele et al. (2012), who fitted single WD models to all spectra in the 20 pc sam- ple. For example, Giammichele et al. (2012) note that the spec- trum of WD0419−487 (RR Caeli) is contaminated by the pres- ence of an M-dwarf companion. As a consequence the WD mass according to Giammichele et al. (2012) is significantly lower (0.22 ± 0.05 M ) than that found by Maxted et al. (2007) (0.440 ± 0.023 M ). Maxted et al. (2007) determined the mass and radius of WD0419−487 from the combined analysis of the radial velocities and the eclipse light curve.

2.1.4. Questionable multiplicity

For eleven WDs, it has been suggested that they are part of a binary or multiple system, however, confirmation or follow-up is lacking. In more detail:

– WD0148 +467 is listed as WD+MS in Holberg et al. (2008b) based on the H ipparcos & Tycho catalogues. We are un- able to find any other objects in these catalogues within two degrees that have a similar parallax and proper motion to WD0148 +467.

– WD0310−688 is suggested to have a second component in the Washington Double Star catalogue. Stau ffer et al. (2010) suggest the companion does not exist.

– Probst (1983) found a possible common proper motion com- panion for WD0341 +182, that is BPM31492.

– Hoard et al. (2007) report a tentative low mass companion for WD0357 +081.

– WD0426 +588 is in a wide binary (Stein2051) with an M-star companion. There is some suggestion that this is a triple sys- tem (Strand 1977). In their model, the red component is an astrometric binary.

– WD0644 +375 is a single WD now, but Ouyed et al. (2011) speculate it used to be a neutron star-WD binary, where the neutron star transitioned to a quark star during a quark nova, enriching the WD with iron, and stripping some of the WD mass. If this is the case, it should be excluded from the com- parison with the BPS models, as in these models the evolu- tion of neutron stars is not taken into account.

– WD0856 +331 was previously identified as being part of a common proper motion binary with HD 77408 (Wegner 1981). However, the magnitudes of the proper motions (Lépine & Shara 2005) and the parallaxes (van Altena et al.

1995; van Leeuwen 2007) di ffer significantly.

– WD1142−645 is listed by Holberg et al. (2008b) as a binary, however, we do not find this to be supported by the associ- ated references or any other literature.

– WD1647 +591 shows possible radial velocity variability for this system (Sa ffer et al. 1998 ), however, as the par- allax and photometric distance agree to within 1.2 sigma (van Leeuwen 2007; Holberg et al. 2008b), we consider it a single WD.

– There is some confusion in the literature as to the multiplic- ity of the system containing WD1917−077. At the time of writing, SIMBAD lists this as a quadruple system. The sup- posed D component appears in the Washington Double Star catalogue, however its proper motion di ffers significantly from the others. The star listed as the C component appears in various literature (Turon et al. 1993; Gould & Chanamé 2004; Lampens et al. 2007) where it is found to have the same proper motion as the A /B component. However, the B /C components were at the time spatially very close lead- ing to blending, which may have impacted their analyses.

Comparison of images between DSS1 and DSS2 surveys show only the A /B components to have any detectable mo- tion between the two epochs laying to rest any suggestion of higher multiplicity.

– Sa ffer et al. (1998) found WD2117 +539 to have possible RV variability, however Foss et al. (1991) did not find variability.

2.1.5. Triples and quadruples

There are a few WDs found in triples and quadruples (Table 2).

The structure of observed multiples tend to be hierarchical, for example triples consist of an inner binary and a distant compan- ion star (Hut & Bahcall 1983). Despite the distance between the companion and the binary, the evolution of these systems can be di fferent from that of isolated binary systems ( Toonen et al.

2016). For example, Thompson (2011) shows that the dynami- cal effect of a third companion on compact DWD binaries can lead to an enhanced rate of mergers and type Ia supernovae. The BPS models presented in this paper do not include the possible interaction of a distant companion. For completeness, we dis- cuss WDs in multiples separately from isolated WDs and bina- ries in the comparison between the synthetic and observed pop- ulations in Sect. 4. Because there are only ∼6 WDs in multiples within 20 pc, including or excluding these systems does not sig- nificantly change our conclusions.

The high-order systems are the following:

– WD0101 +048 is part of a hierarchical quadruple, consisting of a close DWD binary (Maxted et al. 2000) and an MS-MS binary (Caballero 2009). The double MS-binary is a visual binary with a period of ∼29 yr and an angular separation of ∼0.5 mas (Balega et al. 2006). There is some uncertainty regarding the period of the close DWD, however a period of 1.2 d or 6.4 d is most likely (Maxted et al. 2000).

– WD0326−273 is a close DWD (Zuckerman et al. 2003;

Nelemans et al. 2005) with an M 5 star in a wide orbit (Sion & Oswalt 1988; Poveda et al. 1994; Garcés et al.

2011).

– WD0413−077 is part of a resolved WDMS binary, with a K-star companion in a wide orbit (Wegner & McMahan 1988; Tokovinin 2008).

– WD0433 +270 is the outer companion of a spectro- scopic binary of spectral type K2 (Tokovinin et al. 2006;

Zhao et al. 2011; Holberg et al. 2013). The K-binary may also have a planetary mass companion at 0.025

⁰⁰

separation (Lucas & Roche 2002; Holberg et al. 2013).

– WD0727 +482 is in a quadruple system. This system con- sists of a resolved DWD, and an unresolved MS-MS bi- nary of spectral type M (Harrington et al. 1981; Probst 1983;

Sion et al. 1991; Andrews et al. 2012; Janson et al. 2014).

– WD0743−336 is the outer star in a triple system (Tokovinin 2012). The inner system, 171 Pup, is an astrometric binary and is resolved with speckle interferometry.

– WD1633 +572 is in a wide orbit around an eclips- ing MS-MS binary of spectral type M (Silvestri et al.

2002; Sion & Oswalt 1988; Poveda et al. 1994; Feiden &

Chaboyer 2014).

– For WD2054−050, see Sect. 2.1.3.

– WD2351−335 is part of a triple system (Scholz et al. 2004;

Farihi et al. 2005). The inner binary is a visual pair consist-

ing of the WD and an M 3.5-star with a separation of 6.6

⁰⁰

.

The outer star is a M 8.5 star in a wide orbit of about 100

⁰⁰

.

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Table 3. Overview of different BPS models.

Model Description Reference

Star formation history BP Star formation rate and space density depends on time and location 1 in the Galaxy. SFR peaks at early times, declines afterwards

cSFR Constant space density and SFR for 10 Gyr –

Initial period distribution

Abt Log-uniform 2

Lognormal Lognormal distribution with a mean of 5.03 d 3 Common-envelope phase

γα γ = 1.75, αλ = 2; Preferred for unresolved DWDs 4, 5, 6

αα αλ = 2 4, 6

αα2 αλ = 0.25; Preferred for unresolved WDMS 7, 8, 9

Notes. There are two models for the SFH, two for the period distribution, and three for the CE-phase, giving 12 models in total.

1

Boissier & Prantzos (1999);

²

Abt (1983);

³

Raghavan et al. (2010);

⁴

Nelemans et al. (2000);

⁵

Nelemans et al. (2001b);

⁶

Toonen et al. (2012);

7

Zorotovic et al. (2010);

⁸

Toonen & Nelemans (2013);

⁹

Camacho et al. (2014).

2.1.6. Miscellaneous

WD0939 +071 is not included in our sample, because it was mistakenly classified as a WD (Gianninas et al. 2011;

Giammichele et al. 2012). The star is also known as GR 431 and PG 0939 +072 and is reclassified by Gianninas et al. (2011) to be an MS F-type star. WD0806−661 is included as a single star ignoring its brown dwarf companion (Luhman et al. 2011).

3. Stellar and binary population synthesis 3.1. SeBa - a fast stellar and binary evolution code

We employ the population synthesis code SeBa (Portegies Zwart

& Verbunt 1996; Nelemans et al. 2001b; Toonen et al. 2012;

Toonen & Nelemans 2013 to simulate a large number of single stars and binaries. We use SeBa to evolve stars from the zero-age main sequence (ZAMS) until and including the remnant phase.

At every timestep, processes such as stellar winds, mass trans- fer, angular momentum loss, common envelope, magnetic brak- ing, and gravitational radiation are considered with appropriate recipes. SeBa is incorporated into the Astrophysics Multipur- pose Software Environment, or AMUSE. This is a component library with a homogeneous interface structure and can be down- loaded for free at amusecode.org (Portegies Zwart et al. 2009).

In this paper, we employ 12 BPS models. The BPS models are the 2 × 2 × 3 possible permutations of two models for the SFH (BP & cSFR), two models for the initial period distribu- tion (“Abt” & “Lognormal”), and three models for the common- envelope phase (γα, αα, & αα2). These assumptions a ffect the predicted space densities most compared to other uncertainties regarding the evolution and formation of stars and binaries. The models are explained in detail in the following sections and an overview is given in Table 3.

3.2. The initial stellar population

The initial stellar population is generated on a Monte Carlo based approach, according to appropriate distribution functions.

The initial mass of single stars and of binary primaries are drawn between 0.95–10 M from a Kroupa initial mass function (IMF; Kroupa et al. 1993). Furthermore, solar metallicities are assumed. For binaries, unless specified otherwise, the secondary mass is drawn from a uniform mass ratio distribution between 0 and 1 (Duchêne & Kraus 2013), and the eccentricity from a thermal distribution (Heggie 1975) between 0 and 1. For the orbital period (or equivalently the semi-major axis) distribution,

we adopt two models. For model “Abt”, the orbits are drawn from a power-law distribution with an exponent of −1 (Abt 1983) ranging from 0 to 10

⁶

R . For model “Lognormal”, periods are drawn from a lognormal distribution with a mean of 5.03 days, a dispersion of 2.28 (Raghavan et al. 2010), and a maximum period of 10

¹⁰

d. For solar-type stars, the latter distribution has become the preferred distribution (Duquennoy & Mayor 1991;

Raghavan et al. 2010; Duchêne & Kraus 2013; Tokovinin 2014).

3.3. Initial binary fraction

Observational studies have shown that the binary fraction de- pends on the spectral type of the primary star (e.g. Shatsky &

Tokovinin 2002; Raghavan et al. 2010; Duchêne & Kraus 2013).

Due to the properties of the IMF and SFH, the average WD pro- genitor is a ∼2 M (A-type) star for the WD systems under con- sideration in this paper.

For G- and F-type stars observed binary fractions are 44±2%

(Duchêne & Kraus 2013) and 54 ± 2% (Raghavan et al. 2010, more specifically 50 ± 4% for F6–G2 stars and 41 ± 3% for G2–

K3 stars). Studies of OB-associations have shown binary frac- tions of over 70% for O- and B-type stars (Shatsky & Tokovinin 2002; Kobulnicky & Fryer 2007; Kouwenhoven et al. 2007;

Sana et al. 2012). From the most thorough search for com- panions to A-stars (De Rosa et al. 2014), a binary fraction of 43.6 ± 5.3% is estimated.

In this paper, we assume an initial binary fraction of 50% un- less specified otherwise. If an initial binary fraction f other than 0.5 is shown to be appropriate, the predicted number of systems (see Table 4) can easily be adjusted as follows: the number of binaries and merged systems should be multiplied with the cor- rection factor w

bin

, and the number of single WDs with w

sin

. The correction factors are given by:

w

_sin

= hM

_sin

i + hM

bin

i

hM

_sin

i + hM

bin

i f /(1 − f ) , (1) and

w

bin

= hM

_sin

i + hM

bin

i

hM

sin

i(1 − f )/ f + hM

bin

i , (2)

where hM

sin

i is the average mass of a single star and hM

_bin

i

the average (total) mass of a binary system. Assuming the ini-

tial distributions as described in Sect. 3.2 and the full range

in stellar masses of 0.1–100 M , hM

sin

i = 0.49 M

and

hM

_bin

i = 0.74 M for the period distribution of Abt (1983), and

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Table 4. Number of systems with WDs components within 20 pc, see also Fig. 3.

Observations

Single WDs WDMS DWD

Resolved Unresolved Resolved Unresolved

Observed 96.5 ± 3.0 19.2 ± 0.4 0.5 ± 0.6 2.1 ± 0.3 1.0 ± 0.1 (5.0 ± 0.8)

86% complete 112 22 0.58 2.4 1.2 (5.8)

In multiples – 4.0 ± 0.01 0 1.0 ± 0.0 1.0 ± 0.6 (2.0 ± 0.6 )

BPS models

Single WDs Mergers WDMS DWD

SFH Period distr. CE Resolved Unresolved Resolved Unresolved

BP Abt

γα

126 (3.5)

36 (1.9)

30 (0.8)

2.4 (0.21)

20 (0.63)

8.2 (0.40)

αα 43 (2.1) 2.3 (0.21) 4.0 (0.28)

αα2 50 (2.2) 1.3 (0.16) 2.0 (0.20)

BP Lognormal

γα

126 (3.5)

15 (1.2)

40 (0.9)

2.5 (0.22)

28 (0.75)

8.0 (0.40)

αα 19 (1.4) 2.4 (0.22) 4.0 (0.28)

αα2 28 (1.7) 1.5 (0.17) 2.3 (0.22)

cSFR Abt

γα

89 (0.5)

26 (0.1)

22 (0.23)

1.8 (0.07)

15 (0.06)

6.1 (0.04)

αα 30 (0.1) 1.9 (0.07) 3.1 (0.03)

αα2 38 (0.1) 1.0 (0.05) 1.5 (0.02)

cSFR Lognormal

γα

89 (0.5)

12 (0.05)

29 (0.27)

1.9 (0.07)

21 (0.07)

5.8 (0.04)

αα 14 (0.06) 2.0 (0.07) 3.0 (0.03)

αα2 21 (0.07) 1.2 (0.05) 1.7 (0.02)

Notes. The observed sample is based on Giammichele et al. (2012), but see Sect. 2 for adaptations. For unresolved DWDs, we list two numbers.

The first number represents confirmed DWD systems, whereas the number in brackets represents the number of confirmed plus candidate DWDs.

The third line lists the number of WD systems in triples and quadruples, which are not included in the first line. The evolution of these systems has not been simulated in the BPS models. The di fferent BPS models are described in Sect. 3 and an overview is given in Table 3. The selection e ffects described in Sect. 3.7 have been applied to the BPS models. Single WDs are formed by single stellar evolution and mergers in binaries. As such, for a given BPS model, the sum of the “Single stars” column and the “Mergers” column should be compared with the observed number of single WDs. The statistical errors on the BPS simulations are given in brackets.

hM

_sin

i = 0.52 M and hM

_bin

i = 0.78 M for the lognormal pe- riod distribution.

For a lower limit on the binary fraction of 40%, the correc- tion factors are w

bin

= 0.83 and w

sin

= 1.25 for both period distributions. For an upper limit of 60%, the correction factors are w

bin

= 1.15 and w

sin

= 0.77. The uncertainty in the initial binary fraction therefore induces an error on the BPS results of about 15–25%

3.4. Common-envelope evolution

An important phase in the evolution of many binary systems oc- curs when one or both stars fill their Roche lobes, and matter can flow from the donor star through the first Lagrangian point to the companion star. As the evolutionary timescales are shorter for more massive stars, the most massive component of the binary will reach the giant phase first, and is likely to fill its Roche lobe before the companion does. If the mass transfer rate from the donor star increases upon mass loss, a runaway situation ensues, named the common-envelope (CE) phase (Paczynski 1976). The CE-phase is a short-lived phase in which the envelope of the donor star engulfs the companion star. If sufficient energy and angular momentum is transferred to the envelope, it can be ex- pelled, and a merger of the binary can be avoided. The CE-phase plays an essential role in binary star evolution, in particular, in the formation of short-period systems. The orbital outcome is

one of the aspects of binary evolution that a ffects the synthetic binary populations most (e.g. Toonen & Nelemans 2013). De- spite its importance and the enormous e fforts of the community, the CE-phase is not understood in detail.

The classical model for the CE-phase is the α-formalism, which is based on the energy budget (Tutukov & Yungelson 1979). The α-parameter describes the e fficiency with which or- bital energy is consumed to unbind the CE according to

E

gr

= α(E

orb,init

− E

orb,final

), (3) where E

orb

is the orbital energy and E

gr

is the binding energy of the envelope. The orbital and binding energy are as defined in Webbink (1984), where E

gr

is approximated by

E

gr

= GM

d

M

d,env

λR , (4)

with M

_d

the donor mass, M

_d,env

the envelope mass of the donor star, λ the envelope-structure parameter, and R the radius of the donor star. Due to the uncertainty in the value of both α and λ, they are often combined into one parameter αλ.

An alternative method for CE-evolution, is the γ-formalism (Nelemans et al. 2000), which is based on angular momentum balance. The γ-parameter describes the e fficiency with which or- bital angular momentum is used to expel the CE according to

J

_b,init

− J

_b,final

J

b,init

= γ ∆M

d

M

d

+ M

a

, (5)

(8)

where J

b,init

and J

b,final

are the orbital angular momentum of the pre- and post-mass transfer binary respectively, and M

a

is the mass of the companion. The motivation for the γ-formalism comes from the observed mass-ratio distribution of DWD sys- tems that could not be explained by the α-formalism nor stable mass transfer for a Hertzsprung gap donor star (see Nelemans et al. 2000). The idea is that angular momentum can be used for the expulsion of the envelope when there is a large amount of angular momentum available, such as in binaries with similar-mass objects. However, the physical mechanism remains unclear. Interestingly, Woods et al. (2012, see also Woods et al.

2010) suggested an alternative model to create double WDs. This evolutionary path involves stable, non-conservative mass trans- fer between a red giant and an MS star. The e ffect on the orbit is a modest widening with a result alike to the γ-description. Fur- ther studies have to take place to see if this path su ffices to create a significant number of DWDs.

In this paper, we adopt three distinct binary evolution models that di ffer in their treatment of the CE-phase. The models are based on di fferent combinations of the α- and γ-formalism with di fferent values of αλ and γ (see Table 3). In detail:

– In model αα, the α-formalism is used to determine the out- come of every CE-phase. The value of the αλ-parameter (αλ = 2) is based on Nelemans et al. (2000), who deduced this value from reconstructing the second phase of mass transfer for observed DWDs.

– For model γα, the γ-prescription is applied unless the binary contains a compact object or the CE is triggered by a tidal instability rather than dynamically unstable Roche lobe over- flow (see Toonen et al. 2012). The value of the αλ-parameter is equal to that in model αα. The value of the γ-parameter (γ = 1.75) is based on modelling the first phase of mass transfer of observed DWDs (Nelemans et al. 2000).

– Model αα2 is similar to model αα, but with a low value of αλ (αλ = 0.25), such that the binary orbit shrinks more strongly during the CE-phase. The motivation for model αα2 comes from the population of close WDMS, that is post-common envelope binaries. With various tech- niques Zorotovic et al. (2010), Toonen & Nelemans (2013), and Camacho et al. (2014) have shown that the common- envelope phase proceeds less e fficiently than is typically as- sumed in these systems, implying a smaller value for αλ.

This finding is based on the concentration of the observed period-distribution at short periods ranging from a few hours to a few days, but a lack of systems at longer periods (e.g.

Nebot Gómez-Morán et al. 2011).

3.5. Star formation history

Regarding the assumptions about the Galaxy, two models are adopted that di ffer in their treatment of the SFH. This comprises the formation rate of the stars and their assigned positions in the Milky Way.

Model BP is taken from Toonen & Nelemans (2013, based on Nelemans et al. 2004). In this model the star forma- tion rate is a function of time and position in the Galaxy (Boissier & Prantzos 1999). It peaks early in the history of the Galaxy and has decreased substantially since then. We assume the Galactic scale height of our binary systems to be 300 pc (Roelofs et al. 2007a,b). The Galactic star formation rate as a function of time (averaged over space) is shown in the left panel of Fig. 2 in Nelemans et al. (2004). For this project, only the star formation rate in the solar neighbourhood is relevant

Fig. 1. Star formation rate as a function of time for model BP and model cSFR. Regarding model BP, the star formation rate at a Galactocentric distance of 8.5 kpc is shown. To convert the local star formation rate of model cSFR to M

Gyr

⁻¹

pc

⁻²

, a Galactic scale height of 300 pc is assumed (Roelofs et al. 2007b,a).

which is shown in Fig. 1. It peaks around 8 Gyr, and extends to 13.5 Gyr, which Boissier & Prantzos (1999) assume is the age of the Galactic disk. However, from MS and WD pop- ulations, it has been shown that oldest stars within the disk have an age of 8–10 Gyr (e.g. Oswalt et al. 1996; Bergeron et al.

1997; del Peloso et al. 2005; Salaris 2009; Haywood et al. 2013;

Gianninas et al. 2015).

Model cSFR is a more simplistic model of the Milky Way with a constant star formation rate and a homogeneous spa- tial distribution of stellar systems in the solar neighbourhood.

The star formation rate is normalized, such that the total stellar mass in the Galaxy (in the full mass range of 0.1–100 M

) is 6 × 10

¹⁰

M . The spatial distribution is normalized in such way that a spherical region of radius x centred on the Sun contains a fraction of systems in the Galaxy equal to (4πx

³

)/(3V), where V is the Galactic volume of 5 × 10

¹¹

pc

³

. We note that from a more elaborate model distribution of stars in the Galaxy, which is de- pendent on the Galactocentric distance, Nelemans et al. (2001b) found a similar relation between the local space density and the total number of stars in the Galaxy (their Eq. (3)), that is, V = 4.8 × 10

¹¹

pc

³

. For model cSFR, we assume star formation has proceeded for the last 10 Gyr. This time span is appropri- ate for the thin disk, where the majority of objects in the 20 pc sample are located (Sion et al. 2014). The average star formation rate (SFR) in mode cSFR is 6 M yr

⁻¹

(see also Fig. 1).

3.6. Magnitudes

The absolute magnitudes (bolometric, as well as ugriz-bands)

are taken from the WD cooling curves of pure hydrogen atmo-

sphere models (Holberg & Bergeron 2006; Kowalski & Saumon

2006; Tremblay et al. 2011, and references therein). For

MS stars we adopt the absolute ugriz-magnitudes as given by

Kraus & Hillenbrand (2007). For both the MS stars and WDs,

we linearly interpolate between the brightness models. For those

stars that are not included in the grids of brightness models,

the closest gridpoint is taken. V-band magnitudes are calculated

(9)

as a transformation from the g- and r-magnitude according to Jester et al. (2005) for stars.

3.7. Types of white dwarf systems

In this paper we consider six types of stellar systems containing WDs:

– Single star: a star that begins and ends its life as a single star.

– Merger: a single WD that has formed as a result of a merger in a binary system.

– Resolved WDMS: a binary consisting of a WD and a main- sequence (MS) component in a wide orbit. We assume an orbit can be resolved if the angular separation is larger than the critical angular separation s

_crit

:

log(s

crit

) = 0.04556|∆V| − 0.0416, (6) where ∆V is the difference in the V-band magnitude of the two stellar components of the binary and s

crit

in arcseconds.

The critical angular separation is an empirical limit that takes into account the brightness contrast between the stars. It is a fit through the three most compact, resolved binaries (Fig. 2) in our sample of WDMS and DWDs within 20 pc.

For our standard model we exclude the multiple system WD0727 +482 at 0.656

⁰⁰

, as this system is only marginally resolved (Strand et al. 1976). For our optimistic and pes- simistic scenario of resolving binaries, we translate the criti- cal separation to

log(s

crit,opt

) = 0.04556|∆V| − 0.1968, (7) such that a binary similar to WD0727 +482 would just be resolved in our data, and

log(s

crit,pes

) = 0.04556|∆V| + 0.3010, (8) such that a binary with ∆V = 0 is resolved only if the angular separation exceeds 2

⁰⁰

.

– Unresolved WDMS: A binary consisting of a WD and an MS in an orbit with an angular separation less than s

crit

.

This population contains binaries that have undergone a phase of mass transfer (such as post-common-envelope bina- ries) as well as systems in which no mass transfer has taken place. The observed sample of WDMS is strongly affected by selection e ffects. We assume that unresolved WDMS can only be observed as a WDMS when both components are visible, that is, when

∆g ≡ g

WD

− g

_MS

< 1, (9)

and

∆z ≡ z

WD

− z

_MS

> −1, (10)

where g and z represent the magnitudes in the Sloan g- and z-bands of the WD and MS component. We note that in this paper the term “unresolved WDMS” refers to an unresolved WDMS in which both components are visible, unless stated differently.

– Resolved DWD: a binary consisting of two WDs in an orbit with an angular separation larger than s

_crit

. These binaries are all su fficiently wide such that mass transfer does not take place at any point in their evolution.

10⁰ 10¹ 10² 10³

angular separation (") 0

2 4 6 8 10 12

∆V

WDMS DWD Multiple

Fig. 2. V-band magnitude difference as a function of angular separa- tion for the resolved orbits of WDs in Tables 1 and 2. Resolved WDMS are shown with blue circles and DWDs with green squares. The re- solved orbits in triples and quadruples are shown with red diamonds.

The resolved orbits in multiples mainly consist of a WD with an MS- companion (see Table 2). Overplotted are our empirical estimates of the critical angular separation s

crit

. Our standard model of Eq. (6) is shown as the black solid line, our optimistic model as the grey dashed- dotted line (Eq. (7)), and our pessimistic model as the grey dashed line (Eq. (8)).

– Unresolved DWD: a binary consisting of two WDs in an or- bit with an angular separation less than s

crit

. We assume an unresolved DWD can be distinguished from a single WD if both stars contribute significantly to the light, that is, when

∆r ≡ |r

WD1

− r

_WD2

| < 1, (11)

where r represents the magnitudes in the Sloan r-band of each of the WD components (WD component 1 and 2). As for unresolved WDMS, the term “unresolved DWDs” is used in this paper for those unresolved DWDs where both compo- nents contribute to the light, unless stated di fferently.

Other types of WD binaries are not taken into account in this project, such as binaries that are currently interacting (e.g. cata- clysmic variables or AM CVn systems) or binaries with evolved stars, neutron stars, or black holes as companions. These sys- tems have not been observed in the solar neighbourhood, and it is likely that they are much less numerous in general than the binaries considered in this paper.

For the synthetic binaries, the angular separation s on the sky is calculated according to

s = a(1 + e

²

/2)

2d , (12)

where a is the semi-major axis, e is the eccentricity of the orbit, and d the distance from us to the binary given by the Galactic model (Sect. 3.5). The time-averaged distance between the two stars for a given orbit is a(1 + e

²

/2). The factor two arises from averaging over all possible orientations on the sky.

4. White dwarfs within 20 pc

Table 4 shows the number of WD systems within 20 pc as pre-

dicted by the BPS approach for di fferent models of the Galaxy,

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di fferent initial period distributions, and different models of common-envelope evolution. The error on the synthetic number of WD systems in Table 4 represents the statistical error in the simulations. It is estimated by the square root of the total number of systems of that stellar type in the simulations. We have sim- ulated multiple realisations of the local WD populations, which reduces the statistical errors of the BPS models. Besides statisti- cal errors, systematic errors originate due to the uncertainties in binary formation and evolution. The systematic errors dominate over the statistical errors in our simulations. For this reason, sta- tistical errors are often omitted in BPS studies; instead di fferent models of binary evolution are compared to gain insight into the systematic errors.

In Table 4, we show the e ffect of different CE-models, but only for merger systems, unresolved WDMS, and unresolved DWDs; as single stars, resolved WDMS and DWDs are not a ffected by binary evolutionary processes. The most common systems are purely single stars, followed by mergers (in a bi- nary leading to a single WD) and resolved WDMS. The pre- dicted population of resolved WDMS is larger than the popula- tion of resolved DWDs, because not all stars will become a WD within a Hubble time. On the other hand, the predicted popula- tion of unresolved WDMS is smaller than the population of un- resolved DWDs. This is because the observational selection ef- fects on WDMS are much stronger than in DWDs (see Sect. 3.7).

In our simulations, 8–19 unresolved WDMS (1 in ∼1.15)

³

and 0.5–2 unresolved DWDs are discarded (1 in 4–5.5) because of the selection e ffects of Eqs. ( 9), (10). Only very few unresolved DWDs are discarded, which means that the WD components of these DWDs tend to have relatively similar brightnesses. We find that this is because the sample is volume-limited instead of magnitude-limited.

For each type of WD system, the observed number of sys- tems within 20 pc is shown in Table 4. This table also gives a first-order correction for the incompleteness of the 20 pc sam- ple, based on the completeness estimate of Holberg et al. (2016) of 86%. Table 4 also lists the number of WD binaries that are part of triples and quadruples.

The observed number of systems within 20 pc is based on Tables 1, 2, and A.1. For each system, we calculate the probabil- ity that the system is within 20 pc with a Monte Carlo approach that takes into account the uncertainty in the distance as given by Col. 3 of Tables 1, 2, and A.1. As a consequence, some systems with a mean distance just outside of 20 pc have a non-zero prob- ability of being within 20 pc. And equally, some systems inside, but close to, the 20 pc boundary have a non-zero chance to fall outside our sample. The number of systems within 20 pc is then estimated by the sum of the probability of each system. The er- rors on the number of systems within 20 pc are based on the same Monte Carlo study. These errors do not include any uncertainty regarding the binarity of the known systems, that is, whether any of the single WDs have an unseen companion or not. Further- more, these errors do not take into account the uncertainty due to low number statistics.

4.1. Single white dwarfs

Single WDs mostly descend from isolated single stars, but can also be formed from binaries in which the stellar components

3

There are three candidates for these systems which have been de- tected based on astrometric perturbations of M-dwarfs (Delfosse et al.

1999; Winters et al. 2017) within 20 pc. The WD companions have not been detected photometrically so far.

merge. Comparing the observations with the combination of the two channels (Fig. 3a), our models predict roughly the same number of WDs (within a factor of 1.8, i.e. 96.1 and 101–176, respectively). Taking into account an 86% completeness level of the observed sample, this factor reduces to 1.6.

The fraction of single WDs from mergers is not insignificant (10–30% of all single WDs). This is consistent with estimates for the halo (van Oirschot et al. 2014). Additionally, this evolu- tionary channel is interesting in the context of magnetic WDs.

A recent hypothesis for strong magnetic fields in single WDs considers a magnetic dynamo generation during a CE-merger in a binary (Tout et al. 2008). The fraction of magnetic WDs amongst all WDs is poorly estimated due to selection effects, but it ranges from 21 ± 8% within 13 pc and 13 ± 4% within 20 pc from Kawka et al. (2007), to 8% from Sion et al. (2014).

This is consistent with the incidence of mergers in our models, but see Briggs et al. (2015) for a more detailed study.

The synthetic number of single WDs is sensitive to the input assumptions of our models. The di fferent models for the SFH a ffect the predicted number of single WDs (excluding mergers) by a factor of 1.4. The number of merged systems is most de- pendent on the initial distribution of periods, and to a lesser de- gree on the physics of the CE-phase. Regarding the former, in the adopted log-normal distribution, fewer binaries are formed with (relatively short) periods that result in mergers as com- pared to model “Abt”. Regarding the latter, when the CE-phase leads to a stronger shrinkage (which increases from model γα, to αα, to αα2), the CE-phase is more likely to lead to a merger of the stellar components.

4.2. Unresolved WDMS

The selection e ffects of unresolved WDMS systems affects the population strongly; only in about 1 of 1–8 systems are both components visible. As a result, our population models predict 1.0–2.5 unresolved WDMS systems to be visible within 20 pc.

The di fferent models for the initial period distribution of the bi- naries and SFH hardly a ffect the number of unresolved WDMS.

Our modelling of the selection e ffects introduces a sys- tematic uncertainty in the synthetic population of WDMS (see Eqs. (9) and (10)). Equation (9) distinguishes WDMS from ap- parent single MS; Eq. (10) distinguishes WDMS from appar- ent single WDs

⁴

. Neither varying the cut between ∆z > 0 and

∆z > −2, nor making a cut in the i-band instead of the z-band sig- nificantly a ffects the number of unresolved WDMS. Varying the cut between ∆g < 0 and ∆g < 2 leads to a decrease of systems by about 25–42% and an increase by about 40–63%, respec- tively. This is in good agreement with the results of Toonen et al.

(2014).

The boundaries that we apply to di fferentiate between re- solved and unresolved binaries (Eqs. (6)–(8)) do not a ffect the number of predicted unresolved WDMS significantly. In the optimistic scenario of Eq. (7), where binaries can be resolved to smaller angular separations then in the standard scenario of Eq. (6), the number of unresolved WDMS decreases by 7–13%.

In the pessimistic scenario in which binaries can be resolved only down to an angular separation of 2

⁰⁰

(Eq. (8)), the number of un- resolved WDMS decreases by 14–31%.

For compact WDMS that have gone through a CE-phase (i.e. post-common envelope binaries or PCEBs), the preferred

4

In most systems the light of the binary is dominated by that of the

MS star, and therefore we ignore those WDMS that appear as single

WDs in the comparison with the observed sample.

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(a) (b)

Fig. 3. Comparison of the known number of WD systems with that of the synthetic models. On the left, the comparisons for single WDs and resolved binaries are shown, on the right for unresolved binaries. The lines represent the observations and the markers the BPS models. The shaded area around the lines represents the statistical error on the observations from the square-root law. The statistical error is larger than the error given in Table 4 based on the distance estimate of individual systems.

CE-model is αα2 (Sect. 3.4). From these models, 1.0–1.5 WDMS systems are predicted within 20 pc, and 0.7–2.5 includ- ing the uncertainty in selection e ffects. This is consistent with the observed number of 0.5 ± 0.6 from Table 4 (see also Fig. 3b).

The number is based on one unresolved WDMS (WD0419−487 or RR Caeli) that is on the edge of 20 pc with d = 20.13±0.55 pc.

Without the distance restriction of the 20 pc sample, the observed lower limit on the space density of PCEBs is (6–

30) × 10

⁻⁶

pc

⁻³

(Schreiber & Gänsicke 2003). In our models the space density of visible, unresolved WDMS with P < 100 d (i.e. PCEBs) is (4.0–16) × 10

⁻⁶

pc

⁻³

. These space densities are calculated in a cylindrical volume with height above the plane of 200 pc and radii of 200 pc and 500 pc centred on the Sun.

We require both stars to contribute to the light according to Eqs. (9) and (10), and the WDMS to be brighter than 20th mag- nitude in the g-band. Furthermore, the space density is only calculated for the BPS models that are based on the SFH of Boissier & Prantzos (1999; model BP

⁵

), as the homogeneous spatial distribution of stars assumed in model cSFR is not valid at large distances from the Galactic plane. In Toonen & Nelemans (2013) the space density of visible PCEBs was simulated us- ing some of the same models as in this paper, that is, based on the SFH of Boissier & Prantzos (1999, model BP) and the initial period distribution from Abt (1983; model “Abt”). De- pending on which volume is averaged over, and whether model γα, αα or αα2 is applied for the CE-phase, the space den- sity that Toonen & Nelemans (2013) find ranges between (4.0–

15) × 10

⁻⁶

pc

⁻³

. Both theoretical space densities are in good agreement with the observed space density of PCEBs.

5

For model BP the space density of systems goes down when one av- erages over a larger volume (further away from the plane of the Galaxy).

4.3. Unresolved DWDs

The models presented in this paper predict '1.5–8 unresolved DWDs within 20 pc. In the 20 pc sample, there is only one confirmed (isolated) unresolved DWD, WD0135−052, which is in agreement with the lower limit of predicted DWD num- bers based on model αα2 (Fig. 3b). Including those WDs that have been classified as DWD candidates (Sect. 2) increases the observed number to 5 ± 1, in good agreement with our models. Besides these DWD candidates, there are five systems (WD0141−675, WD1223−659, WD1632 +177, WD2008−600, and WD2140+207) whose masses are very close to the lower limit from single stellar evolution ( .0.5 M ), which might have an undetected companion. Additionally, there are two confirmed DWDs (WD0101 +048 and WD 0326−273), that are part of higher-order systems, and one DWD candidate with an MS com- panion (WD2054−050). Given the large uncertainty in the to- tal number of unresolved DWDs, it is not possible to place a strong constraint on the BPS models. We can only conclude that the models are consistent with the observed numbers within the uncertainties.